lambda calculus
lambda calculus (λcalculus) A formalism for representing functions and ways of combining functions, invented around 1930 by the logician Alonzo Church. The following are examples of λexpressions:
λx.x denotes the identity function, which simply returns its argument;
λx.c denotes the constant function, which always returns the constant c regardless of its argument;
λx.f(f(x)) denotes the composition of the function f with itself, i.e. the function that, for any argument x, returns f(f(x)).
Much of the power of the notation derives from the ability to represent higherorder functions. For example, λf.λx.f(f(x))
denotes the (higherorder) function that, when applied to a function f, returns the function obtained by composing f with itself.
As well as a notation, the λcalculus comprises rules for reducing λexpressions to equivalent ones. The most important is the rule of βreduction, by which an expression of the form (λx.e_{1})(e_{2})
reduces to e_{1} with all free occurrences of x replaced by e_{2}. For example, (λx.f(λx.x,x))(a)
reduces to f(λx.x,a)
As a second example, involving a functional variable, the expression (λf.f(a))(λx.g)(x,b))
reduces to (λx.g(x,b))(a)
and hence to g(a,b)
In theoretical terms, the formalism of λcalculus can be shown to be equivalent in expressive power to that of Turing machines. It has a special role in the study of programming languages: one can point to its influence on the design of functional languages such as J. McCarthy's LISP; to P. Landin's reduction of Algol 60 to λcalculus, and to D. Scott's construction of a settheory meaning for the full unrestricted λcalculus – a construction that ushered in the theory of domains in the denotational semantics of programming languages.
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